{"paper":{"title":"Estimates of solutions of elliptic equations with a source reaction term involving the product of the function and its gradient","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Laurent Veron (LMPT), Marie-Fran\\c{c}oise Bidaut-Veron (LMPT), Marta Garcia-Huidobro","submitted_at":"2017-11-30T16:14:35Z","abstract_excerpt":"We study local and global properties of positive solutions of $-{\\Delta}u=u^p]{\\left |{\\nabla u}\\right |}^q$ in a domain ${\\Omega}$ of ${\\mathbb R}^N$, in the range $1\\<p+q$, $p\\geq 0$, $0\\leq q\\< 2$. We first prove a local Harnack inequality and nonexistence of positive solutions in ${\\mathbb R}^N$ when $p(N-2)+q(N-1) \\<N$ or in an exterior domain if $p(N-2)+q(N-1)\\<N$ and $0\\leq q\\<1$. Using a direct Bernstein method we obtain a first range of values of $p$ and $q$ in which $u(x)\\leq c({\\mathrm dist\\,}(x,\\partial\\Omega)^{\\frac{q-2}{p+q-1}}$ This holds in particular if $p+q\\<1+\\frac{4}{n-1}$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1711.11489","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}